Subgroups — C₂×#11726836 #13507094

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₈	C₂	1	2	yes	maximal
3	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#89298	C₂	1	2	yes	maximal
5	C₂×#89300	C₂	1	1	yes	maximal, characteristic
6	#11726836	C₂	1	8	yes	maximal, direct product factor
7	C₂²×Dic₂₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₄	—	2	2	no	—
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6392	C₂²	1	2	yes	—
13	C₂×#6390	—	2	2	no	—
14	C₂×Dic₅₆	C₄	1	8	yes	complemented
15	C₂×Dic₅₆	C₂²	1	4	yes	—
16	C₄×Dic₂₈	—	2	4	no	—
17	C₄×Dic₂₈	C₂²	1	8	yes	—
18	C₇×#346	C₂²	1	4	yes	—
19	#89300	C₂²	1	4	yes	—
20	#89298	C₂²	1	8	yes	—
21	C₂×#2176	—	7	1	no	maximal, Sylow
22	C₂²×Dic₁₄	—	2	2	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	—	2	2	no	—
27	C₂×Dic₇⋊C₄	—	4	2	no	—
28	C₂×C₄×Dic₇	—	4	2	no	—
29	C₂×Dic₂₈	—	2	2	no	—
30	C₂×Dic₂₈	—	2	4	no	—
31	C₂×Dic₂₈	C₂×C₄	1	8	yes	—
32	C₂×Dic₂₈	C₂³	1	4	yes	—
33	C₄×Dic₁₄	—	2	8	no	—
34	C₄×Dic₁₄	—	4	4	no	—
35	C₂×C₇×C₁₆	—	2	2	no	—
36	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₇×C₁₆	C₂³	1	2	yes	—
38	C₂×C₇×C₁₆	C₂³	1	2	yes	—
39	C₄×C₇×C₈	D₄	1	2	yes	—
40	C₄×C₇×C₈	C₂³	1	2	yes	—
41	#6392	—	2	2	no	—
42	#6392	C₂³	1	4	yes	—
43	#6390	—	2	8	no	—
44	Dic₅₆	C₂×C₄	1	16	yes	complemented
45	C₂²×Q₃₂	—	7	1	no	—
46	C₂×C₄×Q₁₆	—	7	2	no	—
47	#2176	—	7	8	no	—
48	C₂×#367	—	7	1	no	—
49	C₂×#358	—	7	2	no	—
50	C₂×#346	D₇	1	1	yes	characteristic
51	C₂²×C₄×C₇	D₈	1	2	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₇	—	4	2	no	—
54	C₂²×Dic₇	—	4	2	no	—
55	C₂×Dic₁₄	—	2	8	no	—
56	C₂×Dic₁₄	—	2	4	no	—
57	C₂×Dic₁₄	—	4	2	no	—
58	C₂×Dic₁₄	—	4	4	no	—
59	C₂×C₇×C₈	—	2	2	no	—
60	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₇×C₈	#41	1	2	yes	—
64	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
65	C₄²×C₇	D₈	1	2	yes	—
66	C₄²×C₇	C₂×D₄	1	2	yes	—
67	C₄⋊Dic₇	—	2	4	no	—
68	C₄⋊Dic₇	—	4	2	no	—
69	Dic₇⋊C₄	—	4	8	no	—
70	C₄×Dic₇	—	4	8	no	—
71	Dic₂₈	—	2	8	no	—
72	Dic₂₈	—	2	4	no	—
73	Dic₂₈	C₂²×C₄	1	8	yes	—
74	C₇×C₁₆	—	2	2	no	—
75	C₇×C₁₆	C₂²×C₄	1	4	yes	—
76	C₂²×Q₁₆	—	7	2	no	—
77	C₂×C₄×Q₈	—	14	2	no	—
78	C₂×Q₃₂	—	7	8	no	—
79	C₂×Q₃₂	—	7	4	no	—
80	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
82	C₄×Q₁₆	—	7	8	no	—
83	C₄×Q₁₆	—	14	4	no	—
84	C₂×#107	—	7	2	no	—
85	C₂×#103	—	14	2	no	—
86	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
87	#367	—	7	4	no	—
88	#358	—	7	8	no	—
89	#346	C₂×D₇	1	4	yes	—
90	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇	#135	1	2	yes	—
96	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₇	—	4	4	no	—
98	C₂×Dic₇	—	4	8	no	—
99	C₂×Dic₇	—	4	4	no	—
100	C₂×Dic₇	—	8	4	no	—
101	Dic₁₄	—	2	8	no	—
102	Dic₁₄	—	4	4	no	—
103	Dic₁₄	—	4	8	no	—
104	C₇×C₈	—	2	2	no	—
105	C₇×C₈	C₄×D₄	1	2	yes	—
106	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
107	C₇×C₈	C₂×#41	1	1	yes	characteristic
108	C₂²×Q₈	—	14	2	no	—
109	C₂×Q₁₆	—	7	8	no	—
110	C₂×Q₁₆	—	7	4	no	—
111	C₂×Q₁₆	—	14	2	no	—
112	C₂×Q₁₆	—	14	4	no	—
113	C₂²×C₈	D₂₈	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
115	C₄×Q₈	—	14	8	no	—
116	C₄×Q₈	—	28	4	no	—
117	C₂×C₄⋊C₄	—	14	2	no	—
118	C₂×C₄⋊C₄	—	28	2	no	—
119	C₂×C₄²	—	28	2	no	—
120	C₂×C₄²	D₂₈	1	1	yes	characteristic
121	Q₃₂	—	7	16	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	C₄×D₇	1	4	yes	—
124	C₂×C₁₆	C₂²×D₇	1	2	yes	—
125	C₂×C₁₆	C₂²×D₇	1	2	yes	—
126	#107	—	7	4	no	—
127	#107	—	14	2	no	—
128	#103	—	14	8	no	—
129	C₄×C₈	D₂₈	1	2	yes	—
130	C₄×C₈	C₂²×D₇	1	2	yes	—
131	C₂²×C₇	C₄×D₈	1	2	yes	—
132	C₂²×C₇	#510	1	4	yes	—
133	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	C₄×D₈	1	2	yes	—
136	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₇	C₂×#135	1	1	yes	characteristic
138	Dic₇	—	4	8	no	—
139	Dic₇	—	8	4	no	—
140	C₂×Q₈	—	14	8	no	—
141	C₂×Q₈	—	14	4	no	—
142	C₂×Q₈	—	28	2	no	—
143	C₂×Q₈	—	28	4	no	—
144	C₂²×C₄	—	28	2	no	—
145	C₂²×C₄	—	28	2	no	—
146	C₂²×C₄	D₅₆	1	2	yes	—
147	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
148	Q₁₆	—	7	8	no	—
149	Q₁₆	—	14	8	no	—
150	Q₁₆	—	14	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
153	C₂×C₈	C₂×D₂₈	1	2	yes	—
154	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
155	C₂×C₈	#1157	1	2	yes	—
156	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
157	C₄⋊C₄	—	14	4	no	—
158	C₄⋊C₄	—	28	8	no	—
159	C₄⋊C₄	—	28	2	no	—
160	C₄²	—	28	8	no	—
161	C₄²	D₅₆	1	2	yes	—
162	C₄²	C₂×D₂₈	1	2	yes	—
163	C₁₆	—	2	2	no	—
164	C₁₆	C₂×C₄×D₇	1	4	yes	—
165	C₂×C₇	#2176	1	4	yes	complemented
166	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
167	C₂×C₇	C₂×#510	1	2	yes	—
168	C₂³	C₂×D₅₆	1	1	yes	characteristic
169	Q₈	—	14	8	no	—
170	Q₈	—	28	4	no	—
171	Q₈	—	28	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	28	4	no	—
174	C₂×C₄	—	28	8	no	—
175	C₂×C₄	—	28	4	no	—
176	C₂×C₄	—	56	4	no	—
177	C₂×C₄	C₄×D₂₈	1	2	yes	—
178	C₂×C₄	C₂×D₅₆	1	4	yes	—
179	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
180	C₂×C₄	#6467	1	2	yes	—
181	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₄×D₂₈	1	2	yes	—
184	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
185	C₈	C₂×#1157	1	1	yes	characteristic
186	C₇	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
187	C₂²	C₄×D₅₆	1	2	yes	—
188	C₂²	#89681	1	4	yes	—
189	C₂²	C₂²×D₅₆	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	28	8	no	—
192	C₄	—	56	4	no	—
193	C₄	C₄×D₅₆	1	2	yes	—
194	C₄	C₂×C₄×D₂₈	1	1	yes	characteristic
195	C₄	C₂×#6467	1	1	yes	characteristic
196	C₂	#11726836	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₅₆	1	1	yes	characteristic
198	C₂	C₂×#89681	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator	C₇×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#346	D₇
hypocenter	C₇	C₂×#2176
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. #11726836	C₂	direct (N × H)
14. C₂×Dic₅₆	C₄	semidirect
44. Dic₅₆	C₂×C₄	semidirect
165. C₂×C₇	#2176	semidirect
186. C₇	C₂×#2176	semidirect
196. C₂	#11726836	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₅₆	2
C₂×C₄×Dic₂₈	2
C₂×C₇×#346	2
C₂×#89298	2
C₂×#89300	2
#11726836	2
C₂×#2176	7

Subgroups — C2×#11726836 #13507094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726836 #13507094